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Crank-Nicolson Galerkin Approximations for Logarithmic Klein-Gordon Equation

Crank-Nicolson Galerkin Approximations for Logarithmic Klein-Gordon Equation

Year:    2025

Author:    Fang Chen, Meng Li, Yanmin Zhao

Journal of Computational Mathematics, Vol. 43 (2025), Iss. 3 : pp. 641–672

Abstract

This paper presents three regularized models for the logarithmic Klein-Gordon equation. By using a modified Crank-Nicolson method in time and the Galerkin finite element method (FEM) in space, a fully implicit energy-conservative numerical scheme is constructed for the local energy regularized model that is regarded as the best one among the three regularized models. Then, the cut-off function technique and the time-space error splitting technique are innovatively combined to rigorously analyze the unconditionally optimal and high-accuracy convergence results of the numerical scheme without any coupling condition between the temporal step size and the spatial mesh width. The theoretical framework is uniform for the other two regularized models. Finally, numerical experiments are provided to verify our theoretical results. The analytical techniques in this work are not limited in the FEM, and can be directly extended into other numerical methods. More importantly, this work closes the gap for the unconditional error/stability analysis of the numerical methods for the logarithmic systems in higher dimensional spaces.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.2312-m2023-0185

Journal of Computational Mathematics, Vol. 43 (2025), Iss. 3 : pp. 641–672

Published online:    2025-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    32

Keywords:    Logarithmic Klein-Gordon equation Finite element method Cut-off Error splitting technique Convergence.

Author Details

Fang Chen

Meng Li

Yanmin Zhao